The main subject of this thesis is the Ising field theory, the field theory describing the scaling limit of the two-dimensional Ising model near its critical point. We study the Ising field theory in low- and high-temperature regimes.In the low-temperature regime we address questions related to the mass spectrum of the bound states that exist in the theory. The overall goal is to study analytic properties of the masses taken as functions of the scaling variable in the theory. At this stage, we are primarily interested in the asymptotic series in the coupling parameter. We describe methods used for developing expansions in the coupling parameter when this parameter is sufficiently small. This gives rise to the low-energy and semiclassical series. Methods developed are then applied to a different model, a model that has many things in common with the Ising field theory, the two-dimensional quantum chromodynamics with infinite number of colors, also known as the ‘t Hooft model.In the second part of the thesis we turn to the Ising field theory in the high- temperature regime. Here we study the problem of scattering in a weak magnetic field. We compute the leading perturbative correction to the scattering amplitude for the 2 -> 2 process. This is done first by means of standard methods based on the direct intermediate-state decomposition and dispersion relations. Then we compute the large energy asymptotic of the amplitude for this process directly using a known relation between the Ising field theory at zero magnetic field and classical sinh-Gordon equation. We obtain consistent results that show the logarithmic growth of the amplitude with energy at this order in magnetic field. Going beyond the leading order we argue that at large energies for a sufficiently weak magnetic field the amplitude exhibits a power-like decay.
I discuss particle content of the Ising field theory (the scaling limit of the Ising model in a magnetic field), in particular the evolution of its mass spectrum under the change of the scaling parameter. I consider both real and pure imaginary magnetic field. Here I address the high-temperature regime, where the spectrum of stable particles is relatively simple (there are from one to three particles, depending on the parameter). My goal is to understand analytic continuations of the masses to the domain of the parameter where they no longer exist as the stable particles. I use the natural tool -- the
2→2 elastic scattering amplitude, with its poles associated with the stable particles, virtual and resonance states in a standard manner. Concentrating attention on the "real" poles (those corresponding to stable and virtual states) I propose a scenario on how the pattern of the poles evolves from the integrable point
T=Tc, H=0 to the free particle point
T>Tc, H=0, and then, along the pure imaginary
H, to the Yang-Lee critical point. Waypoints along this evolution path are located using TFFSA data. I also speculate about likely behavior of some of the resonance poles.
Abstract We study the “three particle coupling”
Γ111(ξ) Γ111ξ , in 2dIsing Field Theory in a magnetic field, as the function of the scaling parameterξ:=h/(−m)15/8, wherem∼Tc−Tandh∼Hare scaled deviation from the critical temperature and scaled external field, respectively. The “φ3coupling”
Γ111 Γ111 is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so thatmis negative. We suggest “standard analyticity”:
(Γ111)2 Γ1112 , as the function ofu:=ξ2, is analytic in the whole complexu-plane except for the branch cut from – ∞ to –u0≈ – 0.03585, the latter branching point –u0being associated with the Yang-Lee edge singularity. Under this assumption, the values of
Γ111 Γ111 at any complexuare expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of
Γ111 Γ111 near the Yang-Lee branching point, as well as its known asymptotic atu →+∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field.
The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.
We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.
I Infinite Renormalization of the Hamiltonian Is Necessary.- II Quantum Field Theory Models.- I. The ?22n Model.- Fock space.- Q space.- The Hamiltonian H(g).- Removing the space cutoff.- Lorentz covariance and the Haag-Kastler axioms.- II. The Yukawa Model.- Preliminaries.- First and second order estimates.- Resolvent convergence and self adjointness.- The Heisenberg picture.- III Boson Quantum Field Models.- I. General Results.- Hermite operators.- Gaussian measures and the Schrodinger representation.- Hermite expansions and Fock space.- II. The Solution of Two-Dimensional Boson Models.- The interaction Hamiltonian.- The free Hamiltonian.- Self-adjointness of H(g).- The local algebras and the Lorentz group automorphisms.- IV Boson Quantum Field Models.- III. Further Developments.- Locally normal representations of the observables.- The construction of the physical vaccum.- Formal perturbation theory and models in three space-time dimensions.- V The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- I. Physics of Quantum Field Models.- Five years of models.- From estimates to physics.- Bound states and resonances.- Phase space localization and renormalization.- VI The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- II. The Cluster Expansion.- The main results.- The cluster expansion.- Clustering and analyticity: proof of the main results.- Convergence: the main ideas.- An equation of Kirkwood-Salsburg type.- Covariance operators.- Derivatives of covariance operators.- Gaussian integrals.- Convergence: the proof completed.- VII Particles and Bound States and Progess Toward Unitarity and Scaling.- VIII Critical Problems in Quantum Fields.- IX Existence of Phase Transitions for ?24 Quantum Fields.- X Critical Exponents and Renormalization in the ?4 Scaling Limit.- Formulation of the problem.- The scaling and critical point limits.- Renormalization of the ?2(x) field.- Existence of the scaling limit.- The Josephson inequality.- XI A Tutorial Course in Constructive Field Theory.- e?tH as a functional integral.- Examples.- Applications of the functional integral representation.- Ising, Gaussian and scaling limits.- Main results.- Correlation inequalities.- Absence of even bound states.- Bound on g.- Bound on dm2/d? and particles.- The conjecture ?(6) ? 0.- Cluster expansions.- The region of convergence.- The zeroth order expansion.- The primitive expansion.- Factorization and partial resummation.- Typical applications.